I'm trying to find a relay (perhaps a pulse-timer relay) that energizes a relay's coil upon activation but does NOT begin to "count down" until after the trigger signal is removed. Does anyone know of a relay or circuit that does that or how to modify an existing pulse-timer relay to do so?

Most PTR's I'm finding will energize a coil and start counting down at the same time. The problem with this is that if my trigger lasts 10 seconds but the PTR is set for 2 seconds, the relay coil will only energize for 2 seconds despite the 10 second trigger. Once the trigger is removed nothing happens since the PTR is waiting for a new trigger signal.

Conversely, if I take a negative trigger and go directly to the 555 timer's trigger pin with a 10 second trigger, the relay will energize for the duration of the trigger but de-energizes upon removal of the trigger. In other words, I "lose" the count down.

I've built a circuit for an automotive application. To avoid having the circuit powered for a constant power line, I would like to use a custom PTR to provide power from the vehicle's 'constant' wire to the circuit upon being triggered from the vehicle's ignition or accessory power, and to stay on for a while after the ign/acc is removed. I hope that makes sense.

I guess I should have specified better. I would like to avoid having to power a microcontroller to avoid a current draw. This is the reason I'm using a trigger to energize a relay that provides power to the microcontroller. Once the trigger (that powers the microcontroller via a relay) goes away, the relay should stay on for a duration of time continuing to power to microcontroller until the relay times out.

I just wired up a reed relay with a coil resistance of 1050 ohms and a 1000uF electrolitic capacitor across the coil. Pulsing the coil with power and ground keeps it energized for about 2 seconds. Being that this is a reed relay and I'm planning to use it in an automotive application I'd rather use a standard Bosch style relay. Thanks to you and the link you provided I ended up finding this which incorporates just that.